Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rewriting for Symmetric Monoidal Categories with Commutative (Co)Monoid Structure

Aleksandar Milosavljevic, Robin Piedeleu, Fabio Zanasi

TL;DR

The paper tackles rewriting of string diagrams in symmetric monoidal categories when each object carries a commutative monoid structure, addressing the tension with Frobenius-based approaches. It develops a combinatorial interpretation to model these diagrams as right-monogamous acyclic cospans of labelled hypergraphs and proves an isomorphism between the free coloured prop plus commutative monoid and the hypergraph-based RMACsp construction. It then shows that rewriting modulo commutative monoid structure corresponds to a restricted double-pushout hypergraph rewriting with weak boundary complements and convex matching, establishing soundness and completeness. The work generalises to multisorted theories, dualises to commutative comonoid settings, and suggests avenues for further exploration of confluence and intermediate algebraic structures, with implications for CD-categories and matrix-like semantic models.

Abstract

String diagrams are pictorial representations for morphisms of symmetric monoidal categories. They constitute an intuitive and expressive graphical syntax, which has found application in a very diverse range of fields including concurrency theory, quantum computing, control theory, machine learning, linguistics, and digital circuits. Rewriting theory for string diagrams relies on a combinatorial interpretation as double-pushout rewriting of certain hypergraphs. As previously studied, there is a `tension' in this interpretation: in order to make it sound and complete, we either need to add structure on string diagrams (in particular, Frobenius algebra structure) or pose restrictions on double-pushout rewriting (resulting in 'convex' rewriting). From the string diagram viewpoint, imposing a full Frobenius structure may not always be natural or desirable in applications, which motivates our study of a weaker requirement: commutative monoid structure. In this work we characterise string diagram rewriting modulo commutative monoid equations, via a sound and complete interpretation in a suitable notion of double-pushout rewriting of hypergraphs.

Rewriting for Symmetric Monoidal Categories with Commutative (Co)Monoid Structure

TL;DR

The paper tackles rewriting of string diagrams in symmetric monoidal categories when each object carries a commutative monoid structure, addressing the tension with Frobenius-based approaches. It develops a combinatorial interpretation to model these diagrams as right-monogamous acyclic cospans of labelled hypergraphs and proves an isomorphism between the free coloured prop plus commutative monoid and the hypergraph-based RMACsp construction. It then shows that rewriting modulo commutative monoid structure corresponds to a restricted double-pushout hypergraph rewriting with weak boundary complements and convex matching, establishing soundness and completeness. The work generalises to multisorted theories, dualises to commutative comonoid settings, and suggests avenues for further exploration of confluence and intermediate algebraic structures, with implications for CD-categories and matrix-like semantic models.

Abstract

String diagrams are pictorial representations for morphisms of symmetric monoidal categories. They constitute an intuitive and expressive graphical syntax, which has found application in a very diverse range of fields including concurrency theory, quantum computing, control theory, machine learning, linguistics, and digital circuits. Rewriting theory for string diagrams relies on a combinatorial interpretation as double-pushout rewriting of certain hypergraphs. As previously studied, there is a `tension' in this interpretation: in order to make it sound and complete, we either need to add structure on string diagrams (in particular, Frobenius algebra structure) or pose restrictions on double-pushout rewriting (resulting in 'convex' rewriting). From the string diagram viewpoint, imposing a full Frobenius structure may not always be natural or desirable in applications, which motivates our study of a weaker requirement: commutative monoid structure. In this work we characterise string diagram rewriting modulo commutative monoid equations, via a sound and complete interpretation in a suitable notion of double-pushout rewriting of hypergraphs.
Paper Structure (9 sections, 13 theorems, 35 equations, 2 figures)

This paper contains 9 sections, 13 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Combinatorial Interpretation
  4. Right-monogamous cospans
  5. Commutative monoids and discrete right-monogamous cospans
  6. The general case.
  7. Characterisation of String Diagram Rewriting
  8. Conclusions and Future Work
  9. String diagrams for SMCs

Key Result

Proposition 3.7

Let $u \xrightarrow{} G \xleftarrow{} v$, $v \xrightarrow{} H \xleftarrow{} w$, $v_1 \xrightarrow{} G_1 \xleftarrow{} w_1$ and $v_2 \xrightarrow{} G_2 \xleftarrow{} w_2$ be right-monogamous acyclic cospans in $\mathrm{Csp}_{D}(\mathbf{Hyp}_{\Sigma,\mathcal{C}})$. Then

Figures (2)

  • Figure 1: Laws of symmetric monoidal categories, for morphisms of a $\mathcal{C}$-coloured prop.
  • Figure :

Theorems & Definitions (57)

  • Definition 2.1: Theories
  • Definition 2.2: Props
  • Example 2.3
  • Remark 2.4
  • Example 2.5: Monoids and Functions
  • Definition 2.6: Cospan
  • Definition 2.7: SMC of cospans
  • Definition 2.8
  • Remark 2.9
  • Example 3.1
  • ...and 47 more